My name is Victor DeCaria. I am an Andrew Mellon Predoctoral Fellow and fifth year graduate student at University of Pittsburgh researching numerical analysis and computational fluid dynamics. I expect to defend my dissertation in the summer of 2019.

Variable Order, Variable Stepsize Timestepping

I research computational fluid dynamics and time discretizations for PDEs. My most recent work is on implicit variable stepsize, variable order (VSVO) methods. We developed a method that only solves one nonlinear system solve per timestep of the same complexity as BDF3, but yields three approximations of orders two, three, and four. This is an embedded triplet from which a VSVO method can be constructed. The method is named Multiple Order One Solve Embedded 234 (MOOSE234). MOOSE234 was tested on the Navier-Stokes equations, and was significantly faster in timing tests than adaptive BDF3. Preprint available at https://arxiv.org/abs/1810.06670.

My goal is to increase the accuracy of popular methods through  computationally cheap post-processing so people with codes using these methods can increase accuracy with little additional programming. In https://arxiv.org/abs/1810.06705, we analyzed and tested a time filter on the usual implicit Euler FEM discretization of the Navier-Stokes equations. The resulting method is second order, A-Stable. The incompressible Navier-Stokes equations are

u_t - \nu\Delta u + u \cdot \nabla u + \nabla p = f(t),

\nabla \cdot u = 0.

The new method we considered is (suppressing the spatial discretization)

\frac{u_{n+1}-u_n}{\Delta t} - \nu \Delta u_{n+1} + u_{n+1}\cdot \nabla u_{n+1} + \nabla p_{n+1} = f(t_{n+1}),

\nabla \cdot u_{n+1}=0.

u_{n+1} = u_{n+1} - \frac{1}{3}(u_{n+1} - 2u_n + u_{n-1}).

The first two equations correspond to the standard implicit Euler method. The last line is the time filter which makes the method second order while retaining nonlinear energy stability. It does not require solving another system. In a FEM setting, the time filter simply requires adding the solution vectors of the three previous approximations.